Nuprl Lemma : integer-sqrt-ext
∀x:ℕ. (∃r:ℕ [(((r * r) ≤ x) ∧ x < (r + 1) * (r + 1))])
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
fastexp: i^n,
efficient-exp-ext,
so_apply: x[s1;s2],
natrec: natrec,
genrec: genrec,
genrec-ap: genrec-ap,
integer-sqrt,
integer-nth-root,
div_nat_induction,
rem_bounds_1,
decidable__lt,
decidable__equal_int,
decidable__squash,
decidable__and,
decidable__less_than',
decidable__int_equal,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
any: any x,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ
Lemmas referenced :
integer-sqrt,
lifting-strict-int_eq,
strict4-decide,
lifting-strict-decide,
lifting-strict-less,
cbv_sqequal,
has-value_wf_base,
efficient-exp-ext,
integer-nth-root,
div_nat_induction,
rem_bounds_1,
decidable__lt,
decidable__equal_int,
decidable__squash,
decidable__and,
decidable__less_than',
decidable__int_equal,
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
baseApply,
closedConclusion,
hypothesisEquality,
lambdaFormation,
callbyvalueReduce
Latex:
\mforall{}x:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} x) \mwedge{} x < (r + 1) * (r + 1))])
Date html generated:
2018_05_21-PM-07_52_25
Last ObjectModification:
2018_05_19-PM-04_50_03
Theory : general
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